Winding angle and maximum winding angle of the two-dimensional random walk

1991 ◽  
Vol 28 (4) ◽  
pp. 717-726 ◽  
Author(s):  
Claude Bélisle ◽  
Julian Faraway
Keyword(s):  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Random Walk ◽  
Random Walks ◽  
Computer Simulations ◽  
Asymptotic Distributions ◽  
Two Dimensional ◽  
Exact Distributions ◽  
The Difference ◽  
Winding Angle

Recent results on the winding angle of the ordinary two-dimensional random walk on the integer lattice are reviewed. The difference between the Brownian motion winding angle and the random walk winding angle is discussed. Other functionals of the random walk, such as the maximum winding angle, are also considered and new results on their asymptotic behavior, as the number of steps increases, are presented. Results of computer simulations are presented, indicating how well the asymptotic distributions fit the exact distributions for random walks with 10m steps, for m = 2, 3, 4, 5, 6, 7.

Winding angle and maximum winding angle of the two-dimensional random walk

1991 ◽  
Vol 28 (04) ◽  
pp. 717-726
Author(s):  
Claude Bélisle ◽  
Julian Faraway
Keyword(s):  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Random Walk ◽  
Random Walks ◽  
Computer Simulations ◽  
Asymptotic Distributions ◽  
Two Dimensional ◽  
Exact Distributions ◽  
The Difference ◽  
Winding Angle

Recent results on the winding angle of the ordinary two-dimensional random walk on the integer lattice are reviewed. The difference between the Brownian motion winding angle and the random walk winding angle is discussed. Other functionals of the random walk, such as the maximum winding angle, are also considered and new results on their asymptotic behavior, as the number of steps increases, are presented. Results of computer simulations are presented, indicating how well the asymptotic distributions fit the exact distributions for random walks with 10 m steps, for m = 2, 3, 4, 5, 6, 7.

Deterministic Random Walks on the Two-Dimensional Grid

2009 ◽  
Vol 18 (1-2) ◽  
pp. 123-144 ◽  
Author(s):  
BENJAMIN DOERR ◽  
TOBIAS FRIEDRICH
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Walk Model ◽  
Two Dimensional ◽  
Expected Number ◽  
Fixed Order ◽  
The Difference ◽  
Propp Machine

Jim Propp's rotor–router model is a deterministic analogue of a random walk on a graph. Instead of distributing chips randomly, each vertex serves its neighbours in a fixed order. We analyse the difference between the Propp machine and random walk on the infinite two-dimensional grid. It is known that, apart from a technicality, independent of the starting configuration, at each time the number of chips on each vertex in the Propp model deviates from the expected number of chips in the random walk model by at most a constant. We show that this constant is approximately 7.8 if all vertices serve their neighbours in clockwise or order, and 7.3 otherwise. This result in particular shows that the order in which the neighbours are served makes a difference. Our analysis also reveals a number of further unexpected properties of the two-dimensional Propp machine.

On the number of times where a simple random walk reaches its maximum

1992 ◽  
Vol 29 (02) ◽  
pp. 305-312 ◽  
Author(s):  
W. Katzenbeisser ◽  
W. Panny
Keyword(s):  
Asymptotic Behavior ◽  
Random Walk ◽  
Random Walks ◽  
Simple Random Walk ◽  
Probability And Statistics

Let Qn denote the number of times where a simple random walk reaches its maximum, where the random walk starts at the origin and returns to the origin after 2n steps. Such random walks play an important role in probability and statistics. In this paper the distribution and the moments of Qn , are considered and their asymptotic behavior is studied.

The ruin problem and cover times of asymmetric random walks and Brownian motions

2000 ◽  
Vol 32 (01) ◽  
pp. 177-192 ◽  
Author(s):  
K. S. Chong ◽  
Richard Cowan ◽  
Lars Holst
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Random Walks ◽  
Cover Time ◽  
Brownian Motions ◽  
Brownian Motion With Drift ◽  
And Brownian Motion ◽  
Cover Times ◽  
Ruin Problem

A simple asymmetric random walk on the integers is stopped when its range is of a given length. When and where is it stopped? Analogous questions can be stated for a Brownian motion. Such problems are studied using results for the classical ruin problem, yielding results for the cover time and the range, both for asymmetric random walks and Brownian motion with drift.

On entrance times of a homogeneous N-dimensional random walk: an identity

1988 ◽  
Vol 25 (A) ◽  
pp. 321-333 ◽  
Author(s):  
J. W. Cohen
Keyword(s):  
Performance Evaluation ◽  
Random Walk ◽  
Boundary Value Problem ◽  
Random Walks ◽  
Mathematical Problem ◽  
Lattice Points ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Computer Performance ◽  
Hilbert Type

Present developments in computer performance evaluation require detailed analysis of N-dimensional random walks on the set of lattice points in the first 2N-ant of Recent research has shown that for the two-dimensional case the inherent mathematical problem can often be formulated as a boundary value problem of the Riemann–Hilbert type. The paper is concerned with a derivation and analysis of an identity for the first entrance times distributions into the boundary of such random walks. The identity formulates a relation between these distributions and the zero-tuples of the kernel of the random walk; the kernel contains all the information concerning the structure of the random walk in the interior of its stage space. For the two-dimensional case the identity is resolved and explicit expressions for the entrance times distributions are obtained.

scholarly journals Law of large numbers for the drift of the two-dimensional wreath product

2021 ◽  
Author(s):  
Anna Erschler ◽  
Tianyi Zheng
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Wreath Product ◽  
Law Of Large Numbers ◽  
Abelian Groups ◽  
Limiting Distribution ◽  
Two Dimensional ◽  
Lamplighter Group ◽  
Large Numbers ◽  
Asymptotic Geometry

AbstractWe prove the law of large numbers for the drift of random walks on the two-dimensional lamplighter group, under the assumption that the random walk has finite $$(2+\epsilon )$$ ( 2 + ϵ ) -moment. This result is in contrast with classical examples of abelian groups, where the displacement after n steps, normalised by its mean, does not concentrate, and the limiting distribution of the normalised n-step displacement admits a density whose support is $$[0,\infty )$$ [ 0 , ∞ ) . We study further examples of groups, some with random walks satisfying LLN for drift and other examples where such concentration phenomenon does not hold, and study relation of this property with asymptotic geometry of groups.

scholarly journals Efficient circuits for quantum walks

2010 ◽  
Vol 10 (5&6) ◽  
pp. 420-434
Author(s):  
C.-F. Chiang ◽  
D. Nagaj ◽  
P. Wocjan
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Probability Distributions ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Problem Size ◽  
Integrable Probability ◽  
The Difference ◽  
General Method

We present an efficient general method for realizing a quantum walk operator corresponding to an arbitrary sparse classical random walk. Our approach is based on Grover and Rudolph's method for preparing coherent versions of efficiently integrable probability distributions \cite{GroverRudolph}. This method is intended for use in quantum walk algorithms with polynomial speedups, whose complexity is usually measured in terms of how many times we have to apply a step of a quantum walk \cite{Szegedy}, compared to the number of necessary classical Markov chain steps. We consider a finer notion of complexity including the number of elementary gates it takes to implement each step of the quantum walk with some desired accuracy. The difference in complexity for various implementation approaches is that our method scales linearly in the sparsity parameter and poly-logarithmically with the inverse of the desired precision. The best previously known general methods either scale quadratically in the sparsity parameter, or polynomially in the inverse precision. Our approach is especially relevant for implementing quantum walks corresponding to classical random walks like those used in the classical algorithms for approximating permanents \cite{Vigoda, Vazirani} and sampling from binary contingency tables \cite{Stefankovi}. In those algorithms, the sparsity parameter grows with the problem size, while maintaining high precision is required.

scholarly journals Combining Losing Games into a Winning Game

2019 ◽  
Vol 18 (01) ◽  
pp. 1950003 ◽  
Author(s):  
Bruno Rémillard ◽  
Jean Vaillancourt
Keyword(s):  
Asymptotic Behavior ◽  
Random Walk ◽  
Random Walks ◽  
Random Environment ◽  
Regime Switching ◽  
Random Environments ◽  
Full Characterization ◽  
Parrondo’S Paradox ◽  
Random Subspaces

Parrondo’s paradox is extended to regime switching random walks in random environments. The paradoxical behavior of the resulting random walk is explained by the effect of the random environment. Full characterization of the asymptotic behavior is achieved in terms of the dimensions of some random subspaces occurring in Oseledec’s theorem. The regime switching mechanism gives our models a richer and more complex asymptotic behavior than the simple random walks in random environments appearing in the literature, in terms of transience and recurrence.

scholarly journals Extensions and solutions for nonlinear diffusion equations and random walks

2019 ◽  
Vol 475 (2231) ◽  
pp. 20190432 ◽  
Author(s):  
E. K. Lenzi ◽  
M. K. Lenzi ◽  
H. V. Ribeiro ◽  
L. R. Evangelista
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Random Walks ◽  
Nonlinear Diffusion ◽  
Fractional Derivatives ◽  
Distribution Functions ◽  
Diffusion Equations ◽  
Nonlinear Diffusion Equations

We investigate a connection between random walks and nonlinear diffusion equations within the framework proposed by Einstein to explain the Brownian motion. We show here how to properly modify that framework in order to handle different physical scenarios. We obtain solutions for nonlinear diffusion equations that emerge from the random walk approach and analyse possible connections with a generalized thermostatistics formalism. Finally, we conclude that fractal and fractional derivatives may emerge in the context of nonlinear diffusion equations, depending on the choice of distribution functions related to the spreading of systems.

A coupling proof of weak convergence

1985 ◽  
Vol 22 (2) ◽  
pp. 447-453
Author(s):  
Peter Guttorp ◽  
Reg Kulperger ◽  
Richard Lockhart
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Weak Convergence ◽  
Random Walks ◽  
Continuous Time ◽  
Similar Argument ◽  
Reflected Brownian Motion ◽  
Time Problem ◽  
The Right ◽  
Reflected Random Walk

Weak convergence to reflected Brownian motion is deduced for certain upwardly drifting random walks by coupling them to a simple reflected random walk. The argument is quite elementary, and also gives the right conditions on the drift. A similar argument works for a corresponding continuous-time problem.

Sign in / Sign up

Export Citation Format

Share Document